As noted in [1]}, Conjecture REF  is not valid for complete graphs. The conjecture also places a lot of emphasis on \(\mu _2\) . Lin, Ning and Wu [2]} proved the Bollobás and Nikiforov conjecture for triangle-free graphs. It should be noted that the cycle \(C_7\)  provides an example of a triangle-free graph for which
\(\mu _1^2 + \mu _2^2 +\mu _3^2 > \frac{2m(\omega - 1)}{\omega }.\) 
